We know that a noncommutative vector bundle is a finitely generated projective $A$-module where $A$ is a non commutative $C^{*}$ algebra. In this question we introduce a particular non commutative $C^{*}$ algebra $A$ and a particular $A$- bimodule $M$, see Construction below. Then our question would be :
Is this particular $M$ a finitely generated projective $A$-module ?
If the answer is yes, how can we compute non commutative characteristic classes of such non commutative vector bundle?
Construction:
Let $N$ be a compact smooth Riemannian manifold. By $\lambda^{p}(N)$ we mean the space of all smooth complex valued $(0,p)$ tensors on $M$. Our pre $C^{*}$ algebra is $A=\lambda^{*}(N)=\oplus \lambda^{p}(N)$ which multiplication is the tensor product. We can equip $A$ to a pre-$C^{*}$ algebra structure as follows: If $\alpha \in \lambda^{p}(N)$, we define $\parallel \alpha \parallel= \sup \parallel \alpha_{x}\parallel ,\;\;x\in N$. The later norm is the standard "multilinear norm" for $p$-linear maps on the tangent space $T_{x} N$.Then we take the sup-norm to define a norm on $\lambda^{*}(N)$ as a graded algebra. (Note that when we restrict $A$ and $M$ to $T_{x} N$, that is we consider a linear problem, we obtain an $AF$ algebra,"http://en.wikipedia.org/wiki/Approximately_finite-dimensional_C*-algebra". What is the K groups of this $AF$ algebra?)
Our module $M$ is $M=\Omega^{*}(N)$, the space of differential forms. Now we define the scalar product as follows:
In a local coordinate $\lambda \in A$ and $\alpha \in M$ are a sum of the following objects:
$\lambda=dx_{i_{1}}\otimes \ldots \otimes dx_{i{k}}\;\;\; \alpha=dx_{j_{1}} \wedge \ldots \wedge dx_{j_{l}}$ we define $\lambda.\alpha=dx_{i_{1}}\wedge \ldots \wedge dx_{i{k}} \wedge dx_{j_{1}} \wedge \ldots \wedge dx_{j_{l}}$
This is independent of choosing a particular local coordinate.
Are there some geometric or topological information of M in the cohomology of $A$ with coefficient in $M$?
